Paper F Sec 3 IP

1 Name: __________________________________ ( ) Class: _____ HWA CHONG INSTITUTION Paper F INTEGRATED MATHEMATICS Level : Secondary Three Duration : 2 hours 30 minutes Do not open this booklet until you are told to do so. INSTRUCTIONS TO CANDIDATES Write your name, class and index number on all the work you hand in. Answer all questions on foolscap paper except Question 13 on graph paper provided and Question 15 on the space provided on question paper. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For  , use either your calculator value or 3.142, unless the question requires the answer in terms of  . INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. You are expected to use an electronic calculator to evaluate explicit numerical expressions. The use of a graphing calculator is not permitted. You are reminded of the need for clear presentation in your answers. Omission of essential working will result in a loss of marks. This question paper consists 9 printed pages, including this page.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2. TRIGONOMETRY Mensuration Arc Length = r , where  is in radians Sector area = 21 2 r  , where  is in radians Formulae for 2 0ax bx c   2 4 2 b b acx a    ABC sin sin sin a b c A B C 2 2 2 c 2 cosa b bc A   1 = sin2 ab C

3 1 (a) Simplify   3 2 27 523 x x   [3] (b) Evaluate 9 5 2 4 2 5   without the use of calculator. Leave your answer in exact form. [2] 2 (a) Solve the equation 3 2 0xx   [3] (b) Solve 2 12 7 2xx x   , giving your answers in exact form. [5] 3 (a) Find the range of values of x for which ( 1)(5 ) 3xx   . [3] (b) Find the values of c for which the equation 24 ( 2) 1x c x c    has equal roots. [4] 4 The expression 32ax x bx has remainder 3 when it is divided by 1x . When the expression is divided by 2x , the remainder is 6. Find the remainder when the expression is divided by 3x . [5] 5 (a) Given that  4lg 1.5 lg lgx y x y   , where x and y are both positive, express, in simplest form, y in terms of x. [4] (b) Solve the simultaneous equations 22log 7 log (11 2 ) 1qp   , 3 9(27)pq . [4] [Turn Over

4 6 (a) Find the range of the function 2f : 2 6 7x x x  for the domain 05 x . [4] (b) The function g is defined by 1g: 2 xx x   , 2x . Find 1g , stating its domain clearly. [3] 7 The sketch below shows part of a quadratic graph, 2y ax bx c   , where a, b and c are constants. Points A, B and C have coordinates (0, 9) , (1,0) and (3,0) respectively. The curve passes through the maximum point D and point E, (4, 9) . (i) Evaluate a, b, and c. [3] (ii) Find the coordinates of D. [2] 8 (i) Sketch the graph of 5 2 2yx   , for 05 x , labelling the intercepts clearly. [3] (ii) Using your sketch in part (i) or otherwise, find the range of values of x for which (a) y is negative, [2] (b) 1y  . [2] −2 −1 1 2 3 4 5 6 −15 −10 −5 x y y x  A (0, -9) C (3, 0)  B (1, 0)  D  E 

5 9 The ACCM Steel Pte Ltd is a producer of stainless steel and aluminum containers. Table A shows the production figures for a particular day. Material Capacity Type Quantity Stainless Steel 10 – gallon 500 Stainless Steel 5 – gallon 350 Stainless Steel 1 – gallon 400 Aluminium 10 – gallon 700 Aluminium 5 – gallon 500 Aluminium 1 – gallon 850 Table B shows the amount of material required to produce each type of container. Capacity Type Amount of material in kg 10 – gallon 6.8 5 – gallon 3.6 1 – gallon 1.4 Table C shows the cost for each type of material. Material Cost Stainless Steel $2.20 Aluminium $2.50 (i) Write down a 2 by 3 matrix representing the data in Table A. [1] (ii) Use the data in Table B and matrix multiplication to determine the amount of each type of material used on that day. [2] (iii) Use the data in Table C and matrix multiplication to determine the total cost of the day’s production, giving your answer to the nearest dollars. [2] [Turn Over

6 10 The solution to this question by accurate drawing will not be accepted. The diagram show a kite ABCD with AB = AD and CB = CD. Point A lies on the x-axis, point B is (4,6), point D is (13,9) and the equation of BC is 22yx . Find (i) the gradient of BD, [2] (ii) Show that the equation of AC is 3 33yx  , [3] (iii) the coordinates of A and of C, [4] (iv) the area of the kite ABCD [2] 11 The figure shows a circle with centre C (3,4). The line x = 8 intersects the circle at points B and D. Point A, (9,4) is another point on the circle. (i) Find the equation of the circle. [2] (ii) Show that 1.171BCD radians. [2] (iii) Hence or otherwise, find the area of the shaded sector. [2] y = 2x – 2 C D (13,9) B (4,6) A x y A B C (3,4) D Shaded sector, S x = 8 y x

7 12 (a) (i) Sketch the graph of 3sin 1yx for the domain 02 x  , showing clearly the y-intercept of graph. [3] (ii) State the range of y. [1] (b) Find all the angles of z, between 0 and 360 inclusive which satisfy the equation  tan 18 2z  . [3] 13 Answer the whole of this question on the graph paper provided. The table shows experimental values of two variables x and y. x 1 2 3 4 5 y 2.40 3.15 3.76 4.25 4.67 It is known that 2ln( )y ax b and x > 0. (i) Express this equation in a form suitable for drawing a straight line. [1] (ii) Plot this graph using the given data. [3] (iii) Use your graph to estimate (a) the value of a and b, [3] (b) the value of y when x = 2.5. [2] [Turn Over

8 14 The figure below shows a picture of the Kerið crater lake, a 3,000 year old volcanic crater lake in South Iceland. The actual lake itself is relatively shallow. The water in the crater lake does not drain out but just rise and fall according to changes in the water level. As a result, the crater resembles a window on the groundwater. Jack would like to explore the possibility of having a water paddling zone for tourists during non-winter months. Hence, there is a need to find out the area of this Kerið crater lake. To approximate the area of the lake, he walks around the perimeter of the lake, taking the measurements shown in the illustration. Using the cosine rule on the three triangles shown, find the area of the lake, round off to nearest whole number. [6] 87 m 84 m 26 m 50 m 104 m 55O 80O A B C D E

9 Detach this page and stapled it together with your solutions on foolscap papers. 15 The diagram shows the function y = f(x) over the domain 4  x  2. The function is undefined for all other values of x. On space provided below, sketch the functions given by (i) 1 f ( )2yx , [2] (ii) f (2 ) 6yx . [2] (i) 1 f ( )2yx (ii) f (2 ) 6yx End of Paper −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 −12 −10 −8 −6 −4 −2 2 4 6 8 10 12 14 16 18 20 x y −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 −12 −10 −8 −6 −4 −2 2 4 6 8 10 12 14 16 18 20 x y